Answer
-D is reflexive
-D is symmetric
-D is not transitive
Work Step by Step
-- D is reflexive:
-For D to be reflexive means that for all real numbers x,x D x. But by definition of D, this means that for all real numbers x,xx= x^2 ≥0, which is true.
--D is symmetric:
- For D to be symmetric means that for all real numbers x and y, if x D y then y D x. But by definition of D, this means that for all real numbers x and y, if x y≥0 then y x≥0, which is true by the commutative law of multiplication.
--D is not transitive:
- For D to be transitive means that for all real numbers x, y,and z,if x D y and y D z then x D z.By definition of D, this means that for all real numbers x, y, and z, if x y≥0 and y z≥0 then xz≥0. But this is false: there exist real numbers x, y, and z such that xy≥0 and y z≥0 but x z 0. As a counterexample, let x =1, y =0, and z =−1. Then x D y and y D z because 1·0≥0 and 0·(−1) ≥0. But x D z because 1·(−1) is not greater than equal to 0.
